Dic₃₆₃₉ #28410033

order	14556
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₆₃₉
unambiguous semidirect product	C₃⋊Dic₁₂₁₃

Splittings

derived, Hall	C₃×C₁₂₁₃ ⋊ C₄
Sylow	C₁₂₁₃ ⋊ Dic₃
Sylow	C₃ ⋊ Dic₁₂₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃₆₃₉
commutator, hypocenter	C₃×C₁₂₁₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₂₁₃	C₂
LCS	G ▷ C₃×C₁₂₁₃
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 1220
transitive degree	μ_t(G) = 14556

A minimal faithful permutation representation — 2 generators on 1220 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,518,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545,546,547,548,549,550,551,552,553,554,555,556,557,558,559,560,561,562,563,564,565,566,567,568,569,570,571,572,573,574,575,576,577,578,579,580,581,582,583,584,585,586,587,588,589,590,591,592,593,594,595,596,597,598,599,600,601,602,603,604,605,606,607,608,609,610,611,612,613,614,615,616,617,618,619,620,621,622,623,624,625,626,627,628,629,630,631,632,633,634,635,636,637,638,639,640,641,642,643,644,645,646,647,648,649,650,651,652,653,654,655,656,657,658,659,660,661,662,663,664,665,666,667,668,669,670,671,672,673,674,675,676,677,678,679,680,681,682,683,684,685,686,687,688,689,690,691,692,693,694,695,696,697,698,699,700,701,702,703,704,705,706,707,708,709,710,711,712,713,714,715,716,717,718,719,720,721,722,723,724,725,726,727,728,729,730,731,732,733,734,735,736,737,738,739,740,741,742,743,744,745,746,747,748,749,750,751,752,753,754,755,756,757,758,759,760,761,762,763,764,765,766,767,768,769,770,771,772,773,774,775,776,777,778,779,780,781,782,783,784,785,786,787,788,789,790,791,792,793,794,795,796,797,798,799,800,801,802,803,804,805,806,807,808,809,810,811,812,813,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841,842,843,844,845,846,847,848,849,850,851,852,853,854,855,856,857,858,859,860,861,862,863,864,865,866,867,868,869,870,871,872,873,874,875,876,877,878,879,880,881,882,883,884,885,886,887,888,889,890,891,892,893,894,895,896,897,898,899,900,901,902,903,904,905,906,907,908,909,910,911,912,913,914,915,916,917,918,919,920,921,922,923,924,925,926,927,928,929,930,931,932,933,934,935,936,937,938,939,940,941,942,943,944,945,946,947,948,949,950,951,952,953,954,955,956,957,958,959,960,961,962,963,964,965,966,967,968,969,970,971,972,973,974,975,976,977,978,979,980,981,982,983,984,985,986,987,988,989,990,991,992,993,994,995,996,997,998,999,1000,1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011,1012,1013,1014,1015,1016,1017,1018,1019,1020,1021,1022,1023,1024,1025,1026,1027,1028,1029,1030,1031,1032,1033,1034,1035,1036,1037,1038,1039,1040,1041,1042,1043,1044,1045,1046,1047,1048,1049,1050,1051,1052,1053,1054,1055,1056,1057,1058,1059,1060,1061,1062,1063,1064,1065,1066,1067,1068,1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090,1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108,1109,1110,1111,1112,1113,1114,1115,1116,1117,1118,1119,1120,1121,1122,1123,1124,1125,1126,1127,1128,1129,1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143,1144,1145,1146,1147,1148,1149,1150,1151,1152,1153,1154,1155,1156,1157,1158,1159,1160,1161,1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172,1173,1174,1175,1176,1177,1178,1179,1180,1181,1182,1183,1184,1185,1186,1187,1188,1189,1190,1191,1192,1193,1194,1195,1196,1197,1198,1199,1200,1201,1202,1203,1204,1205,1206,1207,1208,1209,1210,1211,1212,1213,1214,1215,1216,1217,1218,1219,1220);(1,3,2,4)(6,7)(9,1220)(10,1219)(11,1218)(12,1217)(13,1216)(14,1215)(15,1214)(16,1213)(17,1212)(18,1211)(19,1210)(20,1209)(21,1208)(22,1207)(23,1206)(24,1205)(25,1204)(26,1203)(27,1202)(28,1201)(29,1200)(30,1199)(31,1198)(32,1197)(33,1196)(34,1195)(35,1194)(36,1193)(37,1192)(38,1191)(39,1190)(40,1189)(41,1188)(42,1187)(43,1186)(44,1185)(45,1184)(46,1183)(47,1182)(48,1181)(49,1180)(50,1179)(51,1178)(52,1177)(53,1176)(54,1175)(55,1174)(56,1173)(57,1172)(58,1171)(59,1170)(60,1169)(61,1168)(62,1167)(63,1166)(64,1165)(65,1164)(66,1163)(67,1162)(68,1161)(69,1160)(70,1159)(71,1158)(72,1157)(73,1156)(74,1155)(75,1154)(76,1153)(77,1152)(78,1151)(79,1150)(80,1149)(81,1148)(82,1147)(83,1146)(84,1145)(85,1144)(86,1143)(87,1142)(88,1141)(89,1140)(90,1139)(91,1138)(92,1137)(93,1136)(94,1135)(95,1134)(96,1133)(97,1132)(98,1131)(99,1130)(100,1129)(101,1128)(102,1127)(103,1126)(104,1125)(105,1124)(106,1123)(107,1122)(108,1121)(109,1120)(110,1119)(111,1118)(112,1117)(113,1116)(114,1115)(115,1114)(116,1113)(117,1112)(118,1111)(119,1110)(120,1109)(121,1108)(122,1107)(123,1106)(124,1105)(125,1104)(126,1103)(127,1102)(128,1101)(129,1100)(130,1099)(131,1098)(132,1097)(133,1096)(134,1095)(135,1094)(136,1093)(137,1092)(138,1091)(139,1090)(140,1089)(141,1088)(142,1087)(143,1086)(144,1085)(145,1084)(146,1083)(147,1082)(148,1081)(149,1080)(150,1079)(151,1078)(152,1077)(153,1076)(154,1075)(155,1074)(156,1073)(157,1072)(158,1071)(159,1070)(160,1069)(161,1068)(162,1067)(163,1066)(164,1065)(165,1064)(166,1063)(167,1062)(168,1061)(169,1060)(170,1059)(171,1058)(172,1057)(173,1056)(174,1055)(175,1054)(176,1053)(177,1052)(178,1051)(179,1050)(180,1049)(181,1048)(182,1047)(183,1046)(184,1045)(185,1044)(186,1043)(187,1042)(188,1041)(189,1040)(190,1039)(191,1038)(192,1037)(193,1036)(194,1035)(195,1034)(196,1033)(197,1032)(198,1031)(199,1030)(200,1029)(201,1028)(202,1027)(203,1026)(204,1025)(205,1024)(206,1023)(207,1022)(208,1021)(209,1020)(210,1019)(211,1018)(212,1017)(213,1016)(214,1015)(215,1014)(216,1013)(217,1012)(218,1011)(219,1010)(220,1009)(221,1008)(222,1007)(223,1006)(224,1005)(225,1004)(226,1003)(227,1002)(228,1001)(229,1000)(230,999)(231,998)(232,997)(233,996)(234,995)(235,994)(236,993)(237,992)(238,991)(239,990)(240,989)(241,988)(242,987)(243,986)(244,985)(245,984)(246,983)(247,982)(248,981)(249,980)(250,979)(251,978)(252,977)(253,976)(254,975)(255,974)(256,973)(257,972)(258,971)(259,970)(260,969)(261,968)(262,967)(263,966)(264,965)(265,964)(266,963)(267,962)(268,961)(269,960)(270,959)(271,958)(272,957)(273,956)(274,955)(275,954)(276,953)(277,952)(278,951)(279,950)(280,949)(281,948)(282,947)(283,946)(284,945)(285,944)(286,943)(287,942)(288,941)(289,940)(290,939)(291,938)(292,937)(293,936)(294,935)(295,934)(296,933)(297,932)(298,931)(299,930)(300,929)(301,928)(302,927)(303,926)(304,925)(305,924)(306,923)(307,922)(308,921)(309,920)(310,919)(311,918)(312,917)(313,916)(314,915)(315,914)(316,913)(317,912)(318,911)(319,910)(320,909)(321,908)(322,907)(323,906)(324,905)(325,904)(326,903)(327,902)(328,901)(329,900)(330,899)(331,898)(332,897)(333,896)(334,895)(335,894)(336,893)(337,892)(338,891)(339,890)(340,889)(341,888)(342,887)(343,886)(344,885)(345,884)(346,883)(347,882)(348,881)(349,880)(350,879)(351,878)(352,877)(353,876)(354,875)(355,874)(356,873)(357,872)(358,871)(359,870)(360,869)(361,868)(362,867)(363,866)(364,865)(365,864)(366,863)(367,862)(368,861)(369,860)(370,859)(371,858)(372,857)(373,856)(374,855)(375,854)(376,853)(377,852)(378,851)(379,850)(380,849)(381,848)(382,847)(383,846)(384,845)(385,844)(386,843)(387,842)(388,841)(389,840)(390,839)(391,838)(392,837)(393,836)(394,835)(395,834)(396,833)(397,832)(398,831)(399,830)(400,829)(401,828)(402,827)(403,826)(404,825)(405,824)(406,823)(407,822)(408,821)(409,820)(410,819)(411,818)(412,817)(413,816)(414,815)(415,814)(416,813)(417,812)(418,811)(419,810)(420,809)(421,808)(422,807)(423,806)(424,805)(425,804)(426,803)(427,802)(428,801)(429,800)(430,799)(431,798)(432,797)(433,796)(434,795)(435,794)(436,793)(437,792)(438,791)(439,790)(440,789)(441,788)(442,787)(443,786)(444,785)(445,784)(446,783)(447,782)(448,781)(449,780)(450,779)(451,778)(452,777)(453,776)(454,775)(455,774)(456,773)(457,772)(458,771)(459,770)(460,769)(461,768)(462,767)(463,766)(464,765)(465,764)(466,763)(467,762)(468,761)(469,760)(470,759)(471,758)(472,757)(473,756)(474,755)(475,754)(476,753)(477,752)(478,751)(479,750)(480,749)(481,748)(482,747)(483,746)(484,745)(485,744)(486,743)(487,742)(488,741)(489,740)(490,739)(491,738)(492,737)(493,736)(494,735)(495,734)(496,733)(497,732)(498,731)(499,730)(500,729)(501,728)(502,727)(503,726)(504,725)(505,724)(506,723)(507,722)(508,721)(509,720)(510,719)(511,718)(512,717)(513,716)(514,715)(515,714)(516,713)(517,712)(518,711)(519,710)(520,709)(521,708)(522,707)(523,706)(524,705)(525,704)(526,703)(527,702)(528,701)(529,700)(530,699)(531,698)(532,697)(533,696)(534,695)(535,694)(536,693)(537,692)(538,691)(539,690)(540,689)(541,688)(542,687)(543,686)(544,685)(545,684)(546,683)(547,682)(548,681)(549,680)(550,679)(551,678)(552,677)(553,676)(554,675)(555,674)(556,673)(557,672)(558,671)(559,670)(560,669)(561,668)(562,667)(563,666)(564,665)(565,664)(566,663)(567,662)(568,661)(569,660)(570,659)(571,658)(572,657)(573,656)(574,655)(575,654)(576,653)(577,652)(578,651)(579,650)(580,649)(581,648)(582,647)(583,646)(584,645)(585,644)(586,643)(587,642)(588,641)(589,640)(590,639)(591,638)(592,637)(593,636)(594,635)(595,634)(596,633)(597,632)(598,631)(599,630)(600,629)(601,628)(602,627)(603,626)(604,625)(605,624)(606,623)(607,622)(608,621)(609,620)(610,619)(611,618)(612,617)(613,616)(614,615)

Polycyclic presentation:

2,3,1213,2|a^2=d,b^a=b^2,c^a=c^1212

Conjugacy classes & character table

3642 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₂₇₈
maximal normal subgroup of	Dic₇₂₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(14556, 6);

Dic3639 #28410033